The Fuchsian theorem

The Fuchsian theorem concerns a class of differential equations called Fuchsian equations, which have singular points where solutions may behave unpredictably. The theorem states that if these equations meet specific criteria—particularly related to the nature of their singularities—then they admit solutions that can be expanded into convergent power series near those points. This is important because it guarantees the existence and detailed structure of solutions around singularities, helping mathematicians understand complex behaviors in systems modeled by such equations, especially in contexts like geometry, physics, and complex analysis.